Traditional deep Gaussian processes model the data evolution using a discrete hierarchy, whereas differential Gaussian processes (DIFFGPs) represent the evolution as an infinitely deep Gaussian process. However, prior DIFFGP methods often overlook the uncertainty of kernel hyperparameters and assume them to be fixed and time-invariant, failing to leverage the unique synergy between continuous-time models and approximate inference. In this work, we propose a fully Bayesian approach that treats the kernel hyperparameters as random variables and constructs coupled stochastic differential equations (SDEs) to learn their posterior distribution and that of inducing points. By incorporating estimation uncertainty on hyperparameters, our method enhances the model's flexibility and adaptability to complex dynamics. Additionally, our approach provides a time-varying, comprehensive, and realistic posterior approximation through coupling variables using SDE methods. Experimental results demonstrate the advantages of our method over traditional approaches, showcasing its superior performance in terms of flexibility, accuracy, and other metrics. Our work opens up exciting research avenues for advancing Bayesian inference and offers a powerful modeling tool for continuous-time Gaussian processes.

本研究解决了现有的差分高斯过程（DIFFGP）方法在核超参数不确定性处理上的不足。我们提出了一种完全贝叶斯的方法，将核超参数视为随机变量，并构建耦合的随机微分方程（SDE）来学习其后验分布。结果表明，所提方法在灵活性和准确性方面显著优于传统方法，为贝叶斯推断的深入研究提供了新的工具。 

通过随机微分方程进行完全贝叶斯的差分高斯过程